April 19 1. The Equivalence Principle We described general relativity (GR, Einstein's theory of gravity) on Monday. I told you that in GR, objects move through "straight" lines in curved space rather than feeling gravitational forces, but I never described where that came from. It turns out that all of GR can be derived from the EQUIVALENCE PRINCIPLE: Acceleration is equivalent to being at rest in a gravitational field. Imagine that you're sealed in an elevator... a) and the elevator is floating freely through space b) and the elevator is in free-fall down a tall elevator shaft on Earth In both cases, you don't notice any force acting on you. If you let go of a ball next to you, it will float next to you. There is no experiment you can do that can distinguish these two cases. c) and the elevator is being pulled "upwards" (ie. in the direction of the roof) through space at 9.8 m/s/s (=32 feet/s/s) d) and the elevator is sitting still on the surface of the Earth, where the acceleration due to gravity is 9.8 m/s/s In both cases, you will feel the force of the bottom of the elevator pushing up on your feet. If you let go of a ball next to you, it will fall towards the floor at 9.8 m/s/s. There is no experiment you can do that can distinguish these two cases. All of GR can be derived as consequences of this simple idea (and knowing how much mass it takes to generate a given strength of gravity). 2. Hawking Radiation a) particle/anti-particle pairs in the vacuum According to the uncertainty principle, there are pairs of properties that cannot be simultaneously determined well. We've talked about position and velocity (actually momentum) before. Another pair are ENERGY and TIME. The shorter time we spend measuring the energy, the larger the uncertainty in the energy. If we spend a long time measuring the energy, its uncertainty is much smaller. So this means that we can violate conservation of energy over short periods of time! The more we violate it, the less time we're allowed to do it for. Think of a cash register. At any instant, the amount of money in it might be off by quite a bit (eg. if the cashier has a pile of $10 bills in their hand so they can give someone change, or just after they've taken a $20 bill from a customer but haven't given back the change yet). At the end of the day, the amount in bills needs to be about right, but you might need to go get some rolls of coins before you open the next morning. At the end of the year, the IRS (and your shareholders!) will be mighty unhappy if every single penny isn't accounted for unless you're Enron. The amount of energy in empty space is zero (ignoring a few complications). But over short periods of time, it can fluctuate because of the uncertainty principle! The way it does this is by the creation of pairs of particles and anti-particles (eg. electrons and anti-electrons (also known as positrons), neutrinos and anti-neutrinos, protons and anti-protons, photon pairs, etc.). They have a certain amount of time they can exist before they need to come together and annihilate so the accounts can be settled. The higher the energy (eg. the larger the mass), the shorter the time they're allowed to exist. So proton/anti-proton pairs don't go very far, while neutrino/anti-neutrino pairs can travel for a while before they need to annihilate. The effects of these have been experimentally verified... if you try to calculate the energy levels of complicated atoms without taking these fluctuations into account, you get the wrong answer. b) at the event horizon What happens if a particle pair is created just outside the event horizon? One can fall into the black hole within the time its allowed to exist. Once it's past the event horizon, it can't come back out to annihilate with its partner! Uh oh... So what happens to its partner? It keeps going in the opposite direction, and eventually escapes! So from far away, we see particles (usually photons) coming off the event horizon. This is called HAWKING RADIATION, named after the famous British physicist Stephen Hawking who first derived it. Where does the energy come from? The energy of the gravitational field. So the gravitational field must lose energy and get weaker. The only way that can happen is if the mass of the black hole goes down. So the black hole shines with Hawking radiation while slowly losing mass. [Someone asked how this works in detail. Unfortunately, the math gets really horrible, but a quick summary is that the particle that falls through the event horizon has negative energy at the time, and so the black hole absorbs negative energy and loses mass. You can't normally have negative energy, but you can on these short timescales where you're violating conservation of energy. You don't need to understand this - I don't entirely understand it myself.] c) temperature and luminosity The spectrum of Hawking radiation is a perfect blackbody (thermal spectrum). Therefore, it has a temperature. One way of thinking about Hawking radiation is as a tidal stretching of particle/anti-particle pairs... so it's more prominant if the tides near the event horizon are stronger. Smaller black holes have stronger tides, so the temperature is INVERSELY PROPORTIONAL to the mass of the black hole. We can calculate the luminosity from the Stefan-Boltzman law: L = 4 pi R^2 sigma T^4 R is the Schwarzschild radius, and is proportional to the mass M T is the temperature, and is inversely proportional to the mass M so L is proportional to the inverse square of the mass M Bigger black holes shines *less* than smaller ones, and at a lower temperature! Let's look at some numbers for our stereotypical 10 solar mass black hole: T = 10^-8 K (0.00000001K.... that's pretty cold) L = 10^-30 Watts so it would take 100000000000000000000000000000000 of these to be as bright as a 100W light bulb! So we can't use this as a way of detecting black holes... they're much much too faint! d) evaporation I didn't make much of a point of this in class, but it's worth pointing out that because the mass of the black hole shrinks, eventually it will disappear! As you might expect, big black holes last for much longer than small black holes (they have a larger mass to begin with, and they have a much lower luminosity. Think about the argument we made for how long stars of different mass live). A 10 solar mass black hole will last for 10^59 times the current age of the universe before it evaporates! 3. Super-massive black holes (the following has nothing at all to do with stellar evolution, but it does have to do with other astronomical black holes. you are responsible for this material) About 1% of galaxies have very bright points at the center. We call these Active Galactic Nuclei. When they are very far away, the central point outshines the entire rest of the galaxy, which is too faint to see, and is called a QUASAR. Quasars and AGN are often sources of X-rays and radio emission. Some of these points fluctuate very rapidly (on scales of a few hours or more), indicating that they must be quite small. We can weigh the central point by looking at the velocities of stars or gas and dust disks that orbit the central point and using the laws of gravity. We usually come up with a number of between 1 million and 1 billion solar masses, depending on the galaxy. We've done this both for galaxies with and without active nuclei. Every time we've ever done this, we've come up with a number > 1 million solar masses... including for our own Milky Way! The central point in the Milky Way is around 2.6 million solar masses. Let's compare this with our criteria for finding stellar mass black holes: - we have an object with a mass which is much much much too high to be held up by any force we can possibly imagine - the maximum size is so small that it must be a single object that holds all that mass - the gas and stars orbiting around it show signs of being in a deep grav. potential, based on the X-rays and radio emission they give off As with stars, none of these is definitive proof that there are super-massive black holes in the centers of galaxies, and won't be until we can get high- resolution X-ray images of the accretion disk. But we can't even begin to imagine anything else that could be that massive in that small a space. So these are almost definitely black holes.